On Results Of Mixed-valued Logical Control Systems

Boolean control networks are a class of particular discrete dynamical control systems defined on finite set and composed of logical state variables, logical con-trol variables and logical functions, which have important application background in biological genetic regulation networks. In recent years, an important achievement of logical control network theory is the matrix semi-tensor product method. By using the matrix semi-tensor product, a logical function with respect to logical state variables and logical control variables can be rewritten in a bi-linear form. The whole logical dynamical equations can also be written in a matrix semi-tensor product form. In this new theoretical frame, some concepts, methods and problems of traditional con-trol theory can be generalized to logical control networks. For example, the concepts of state space and subspace are successfully extended into logical control networks, which play an important role to system decomposition and disturbance decoupling. However, most of the existing results are for bi-valued logical control networks. For mixed-valued logical control networks, few research work on the state-space theory is reported. There are many differences between mixed-valued logical control networks and bi-valued logical control networks. Mixed-valued logical control network theory has practical background in finite evolutionary games.This thesis investigates some problems of mixed-valued logical control network-s. The obtained results are stated as follows:First, it is obtained a necessary and sufficient condition for complementary subspaces, a testing condition is derived for regular subspaces, and a concrete algorithm to construct complementary subspaces is designed. Second, a criterion for friendly subspace is proposed and an algorithm to compute a friendly subspace is designed. Finally, a concept of invariant subspace without regularity assumption is proposed and some testing conditions are derived.